I think as long as you avoid the classic cliches of the Trainer waking up late and getting the last Pokemon from the Professor, anything will work.
In fact, if I may make a recommendation, try coming up with a creative way for the Trainer to get their starter Pokemon, that doesn't involve them getting it from the Professor. I love Fics that do this, because it really breaks the mold of what's usually seen in a Fic.
But yes, the first-person Trainer Fic can indeed work.
A Winner Is You!!!
If you spend too much time thinking about a thing, you'll never get it done.
- Bruce Lee